LMFDB - Embedded newform 2304.4.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	2304.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.4222 q^{5} +22.4222 q^{7} +O(q^{10})$	$q-18.4222 q^{5} +22.4222 q^{7} -53.6888 q^{11} -7.15559 q^{13} -39.6888 q^{17} +125.689 q^{19} -99.1556 q^{23} +214.378 q^{25} +205.800 q^{29} +147.489 q^{31} -413.066 q^{35} -125.689 q^{37} +506.444 q^{41} -413.689 q^{43} +313.911 q^{47} +159.755 q^{49} -44.3331 q^{53} +989.066 q^{55} +324.000 q^{59} -324.000 q^{61} +131.822 q^{65} -464.266 q^{67} -1052.84 q^{71} +1022.27 q^{73} -1203.82 q^{77} -602.910 q^{79} +15.8217 q^{83} +731.156 q^{85} -381.378 q^{89} -160.444 q^{91} -2315.47 q^{95} +659.154 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{5} + 16 q^{7}+O(q^{10}) $ 2 * q - 8 * q^5 + 16 * q^7	$ 2 q - 8 q^{5} + 16 q^{7} + 8 q^{11} - 72 q^{13} + 36 q^{17} + 136 q^{19} - 256 q^{23} + 198 q^{25} + 152 q^{29} - 80 q^{31} - 480 q^{35} - 136 q^{37} + 436 q^{41} - 712 q^{43} + 224 q^{47} - 142 q^{49} + 344 q^{53} + 1632 q^{55} + 648 q^{59} - 648 q^{61} - 544 q^{65} + 456 q^{67} - 2048 q^{71} + 660 q^{73} - 1600 q^{77} + 496 q^{79} - 776 q^{83} + 1520 q^{85} - 532 q^{89} + 256 q^{91} - 2208 q^{95} - 1220 q^{97}+O(q^{100}) $ 2 * q - 8 * q^5 + 16 * q^7 + 8 * q^11 - 72 * q^13 + 36 * q^17 + 136 * q^19 - 256 * q^23 + 198 * q^25 + 152 * q^29 - 80 * q^31 - 480 * q^35 - 136 * q^37 + 436 * q^41 - 712 * q^43 + 224 * q^47 - 142 * q^49 + 344 * q^53 + 1632 * q^55 + 648 * q^59 - 648 * q^61 - 544 * q^65 + 456 * q^67 - 2048 * q^71 + 660 * q^73 - 1600 * q^77 + 496 * q^79 - 776 * q^83 + 1520 * q^85 - 532 * q^89 + 256 * q^91 - 2208 * q^95 - 1220 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−18.4222	−1.64773	−0.823866	−	0.566785i	$-0.808187\pi$
$5$	−18.4222	−1.64773	−0.823866	+	0.566785i	$0.808187\pi$
$6$	0	0
$7$	22.4222	1.21069	0.605343	−	0.795965i	$-0.293036\pi$
$7$	22.4222	1.21069	0.605343	+	0.795965i	$0.293036\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−53.6888	−1.47162	−0.735809	−	0.677190i	$-0.763198\pi$
$11$	−53.6888	−1.47162	−0.735809	+	0.677190i	$0.763198\pi$
$12$	0	0
$13$	−7.15559	−0.152662	−0.0763309	−	0.997083i	$-0.524321\pi$
$13$	−7.15559	−0.152662	−0.0763309	+	0.997083i	$0.524321\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−39.6888	−0.566233	−0.283116	−	0.959086i	$-0.591368\pi$
$17$	−39.6888	−0.566233	−0.283116	+	0.959086i	$0.591368\pi$
$18$	0	0
$19$	125.689	1.51763	0.758816	−	0.651306i	$-0.225778\pi$
$19$	125.689	1.51763	0.758816	+	0.651306i	$0.225778\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−99.1556	−0.898929	−0.449465	−	0.893298i	$-0.648385\pi$
$23$	−99.1556	−0.898929	−0.449465	+	0.893298i	$0.648385\pi$
$24$	0	0
$25$	214.378	1.71502
$26$	0	0
$27$	0	0
$28$	0	0
$29$	205.800	1.31780	0.658898	−	0.752232i	$-0.271023\pi$
$29$	205.800	1.31780	0.658898	+	0.752232i	$0.271023\pi$
$30$	0	0
$31$	147.489	0.854508	0.427254	−	0.904132i	$-0.359481\pi$
$31$	147.489	0.854508	0.427254	+	0.904132i	$0.359481\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−413.066	−1.99489
$36$	0	0
$37$	−125.689	−0.558463	−0.279231	−	0.960224i	$-0.590080\pi$
$37$	−125.689	−0.558463	−0.279231	+	0.960224i	$0.590080\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	506.444	1.92910	0.964552	−	0.263892i	$-0.0850063\pi$
$41$	506.444	1.92910	0.964552	+	0.263892i	$0.0850063\pi$
$42$	0	0
$43$	−413.689	−1.46714	−0.733569	−	0.679615i	$-0.762147\pi$
$43$	−413.689	−1.46714	−0.733569	+	0.679615i	$0.762147\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	313.911	0.974226	0.487113	−	0.873339i	$-0.338050\pi$
$47$	313.911	0.974226	0.487113	+	0.873339i	$0.338050\pi$
$48$	0	0
$49$	159.755	0.465759
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−44.3331	−0.114898	−0.0574492	−	0.998348i	$-0.518297\pi$
$53$	−44.3331	−0.114898	−0.0574492	+	0.998348i	$0.518297\pi$
$54$	0	0
$55$	989.066	2.42483
$56$	0	0
$57$	0	0
$58$	0	0
$59$	324.000	0.714936	0.357468	−	0.933925i	$-0.383640\pi$
$59$	324.000	0.714936	0.357468	+	0.933925i	$0.383640\pi$
$60$	0	0
$61$	−324.000	−0.680065	−0.340032	−	0.940414i	$-0.610438\pi$
$61$	−324.000	−0.680065	−0.340032	+	0.940414i	$0.610438\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	131.822	0.251546
$66$	0	0
$67$	−464.266	−0.846554	−0.423277	−	0.906000i	$-0.639120\pi$
$67$	−464.266	−0.846554	−0.423277	+	0.906000i	$0.639120\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1052.84	−1.75985	−0.879927	−	0.475109i	$-0.842409\pi$
$71$	−1052.84	−1.75985	−0.879927	+	0.475109i	$0.842409\pi$
$72$	0	0
$73$	1022.27	1.63900	0.819501	−	0.573078i	$-0.194251\pi$
$73$	1022.27	1.63900	0.819501	+	0.573078i	$0.194251\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1203.82	−1.78167
$78$	0	0
$79$	−602.910	−0.858642	−0.429321	−	0.903152i	$-0.641247\pi$
$79$	−602.910	−0.858642	−0.429321	+	0.903152i	$0.641247\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.8217	0.0209236	0.0104618	−	0.999945i	$-0.496670\pi$
$83$	15.8217	0.0209236	0.0104618	+	0.999945i	$0.496670\pi$
$84$	0	0
$85$	731.156	0.933000
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−381.378	−0.454224	−0.227112	−	0.973869i	$-0.572928\pi$
$89$	−381.378	−0.454224	−0.227112	+	0.973869i	$0.572928\pi$
$90$	0	0
$91$	−160.444	−0.184825
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2315.47	−2.50065
$96$	0	0
$97$	659.154	0.689969	0.344984	−	0.938608i	$-0.387884\pi$
$97$	659.154	0.689969	0.344984	+	0.938608i	$0.387884\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	498.510	0.491125	0.245562	−	0.969381i	$-0.421027\pi$
$101$	498.510	0.491125	0.245562	+	0.969381i	$0.421027\pi$
$102$	0	0
$103$	−196.821	−0.188285	−0.0941425	−	0.995559i	$-0.530011\pi$
$103$	−196.821	−0.188285	−0.0941425	+	0.995559i	$0.530011\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	359.378	0.324695	0.162347	−	0.986734i	$-0.448093\pi$
$107$	359.378	0.324695	0.162347	+	0.986734i	$0.448093\pi$
$108$	0	0
$109$	−1969.73	−1.73088	−0.865441	−	0.501011i	$-0.832962\pi$
$109$	−1969.73	−1.73088	−0.865441	+	0.501011i	$0.832962\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	693.643	0.577456	0.288728	−	0.957411i	$-0.406768\pi$
$113$	693.643	0.577456	0.288728	+	0.957411i	$0.406768\pi$
$114$	0	0
$115$	1826.66	1.48119
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−889.911	−0.685529
$120$	0	0
$121$	1551.49	1.16566
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1646.53	−1.17816
$126$	0	0
$127$	2656.78	1.85631	0.928153	−	0.372199i	$-0.121396\pi$
$127$	2656.78	1.85631	0.928153	+	0.372199i	$0.121396\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	615.734	0.410664	0.205332	−	0.978692i	$-0.434173\pi$
$131$	615.734	0.410664	0.205332	+	0.978692i	$0.434173\pi$
$132$	0	0
$133$	2818.22	1.83737
$134$	0	0
$135$	0	0
$136$	0	0
$137$	613.290	0.382459	0.191230	−	0.981545i	$-0.438753\pi$
$137$	613.290	0.382459	0.191230	+	0.981545i	$0.438753\pi$
$138$	0	0
$139$	1899.29	1.15896	0.579480	−	0.814987i	$-0.303256\pi$
$139$	1899.29	1.15896	0.579480	+	0.814987i	$0.303256\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	384.175	0.224660
$144$	0	0
$145$	−3791.29	−2.17137
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−976.377	−0.536832	−0.268416	−	0.963303i	$-0.586500\pi$
$149$	−976.377	−0.536832	−0.268416	+	0.963303i	$0.586500\pi$
$150$	0	0
$151$	683.132	0.368162	0.184081	−	0.982911i	$-0.441069\pi$
$151$	683.132	0.368162	0.184081	+	0.982911i	$0.441069\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2717.07	−1.40800
$156$	0	0
$157$	511.109	0.259815	0.129907	−	0.991526i	$-0.458532\pi$
$157$	511.109	0.259815	0.129907	+	0.991526i	$0.458532\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2223.29	−1.08832
$162$	0	0
$163$	−2425.95	−1.16574	−0.582869	−	0.812566i	$-0.698070\pi$
$163$	−2425.95	−1.16574	−0.582869	+	0.812566i	$0.698070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−337.332	−0.156309	−0.0781544	−	0.996941i	$-0.524903\pi$
$167$	−337.332	−0.156309	−0.0781544	+	0.996941i	$0.524903\pi$
$168$	0	0
$169$	−2145.80	−0.976694
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2648.29	−1.16385	−0.581924	−	0.813243i	$-0.697700\pi$
$173$	−2648.29	−1.16385	−0.581924	+	0.813243i	$0.697700\pi$
$174$	0	0
$175$	4806.82	2.07635
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2907.29	1.21397	0.606986	−	0.794713i	$-0.292379\pi$
$179$	2907.29	1.21397	0.606986	+	0.794713i	$0.292379\pi$
$180$	0	0
$181$	−3682.80	−1.51238	−0.756188	−	0.654354i	$-0.772940\pi$
$181$	−3682.80	−1.51238	−0.756188	+	0.654354i	$0.772940\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2315.47	0.920197
$186$	0	0
$187$	2130.85	0.833277
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1279.56	−0.484740	−0.242370	−	0.970184i	$-0.577925\pi$
$191$	−1279.56	−0.484740	−0.242370	+	0.970184i	$0.577925\pi$
$192$	0	0
$193$	−4836.84	−1.80396	−0.901978	−	0.431783i	$-0.857885\pi$
$193$	−4836.84	−1.80396	−0.901978	+	0.431783i	$0.857885\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2869.31	−1.03771	−0.518857	−	0.854861i	$-0.673642\pi$
$197$	−2869.31	−1.03771	−0.518857	+	0.854861i	$0.673642\pi$
$198$	0	0
$199$	652.242	0.232343	0.116171	−	0.993229i	$-0.462938\pi$
$199$	652.242	0.232343	0.116171	+	0.993229i	$0.462938\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4614.49	1.59544
$204$	0	0
$205$	−9329.82	−3.17865
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6748.08	−2.23337
$210$	0	0
$211$	−537.511	−0.175373	−0.0876866	−	0.996148i	$-0.527947\pi$
$211$	−537.511	−0.175373	−0.0876866	+	0.996148i	$0.527947\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7621.06	2.41745
$216$	0	0
$217$	3307.02	1.03454
$218$	0	0
$219$	0	0
$220$	0	0
$221$	283.997	0.0864421
$222$	0	0
$223$	−4041.98	−1.21377	−0.606885	−	0.794790i	$-0.707581\pi$
$223$	−4041.98	−1.21377	−0.606885	+	0.794790i	$0.707581\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3070.22	0.897699	0.448850	−	0.893607i	$-0.351834\pi$
$227$	3070.22	0.897699	0.448850	+	0.893607i	$0.351834\pi$
$228$	0	0
$229$	205.110	0.0591881	0.0295940	−	0.999562i	$-0.490579\pi$
$229$	205.110	0.0591881	0.0295940	+	0.999562i	$0.490579\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.3776	−0.00376137	−0.00188068	−	0.999998i	$-0.500599\pi$
$233$	−13.3776	−0.00376137	−0.00188068	+	0.999998i	$0.500599\pi$
$234$	0	0
$235$	−5782.93	−1.60526
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4327.99	−1.17136	−0.585679	−	0.810543i	$-0.699172\pi$
$239$	−4327.99	−1.17136	−0.585679	+	0.810543i	$0.699172\pi$
$240$	0	0
$241$	−1508.31	−0.403150	−0.201575	−	0.979473i	$-0.564606\pi$
$241$	−1508.31	−0.403150	−0.201575	+	0.979473i	$0.564606\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2943.04	−0.767446
$246$	0	0
$247$	−899.378	−0.231684
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4871.47	−1.22504	−0.612518	−	0.790456i	$-0.709843\pi$
$251$	−4871.47	−1.22504	−0.612518	+	0.790456i	$0.709843\pi$
$252$	0	0
$253$	5323.55	1.32288
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1665.55	0.404258	0.202129	−	0.979359i	$-0.435214\pi$
$257$	1665.55	0.404258	0.202129	+	0.979359i	$0.435214\pi$
$258$	0	0
$259$	−2818.22	−0.676122
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7167.64	−1.68052	−0.840258	−	0.542188i	$-0.817596\pi$
$263$	−7167.64	−1.68052	−0.840258	+	0.542188i	$0.817596\pi$
$264$	0	0
$265$	816.713	0.189322
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5453.84	−1.23616	−0.618079	−	0.786116i	$-0.712089\pi$
$269$	−5453.84	−1.23616	−0.618079	+	0.786116i	$0.712089\pi$
$270$	0	0
$271$	−5416.20	−1.21406	−0.607031	−	0.794678i	$-0.707640\pi$
$271$	−5416.20	−1.21406	−0.607031	+	0.794678i	$0.707640\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11509.7	−2.52385
$276$	0	0
$277$	2648.75	0.574542	0.287271	−	0.957849i	$-0.407252\pi$
$277$	2648.75	0.574542	0.287271	+	0.957849i	$0.407252\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6664.57	−1.41486	−0.707429	−	0.706784i	$-0.750145\pi$
$281$	−6664.57	−1.41486	−0.707429	+	0.706784i	$0.750145\pi$
$282$	0	0
$283$	5630.84	1.18275	0.591376	−	0.806396i	$-0.298585\pi$
$283$	5630.84	1.18275	0.591376	+	0.806396i	$0.298585\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11355.6	2.33554
$288$	0	0
$289$	−3337.80	−0.679381
$290$	0	0
$291$	0	0
$292$	0	0
$293$	908.374	0.181119	0.0905593	−	0.995891i	$-0.471135\pi$
$293$	908.374	0.181119	0.0905593	+	0.995891i	$0.471135\pi$
$294$	0	0
$295$	−5968.79	−1.17802
$296$	0	0
$297$	0	0
$298$	0	0
$299$	709.517	0.137232
$300$	0	0
$301$	−9275.82	−1.77624
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5968.79	1.12056
$306$	0	0
$307$	414.671	0.0770896	0.0385448	−	0.999257i	$-0.487728\pi$
$307$	414.671	0.0770896	0.0385448	+	0.999257i	$0.487728\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1615.91	−0.294629	−0.147315	−	0.989090i	$-0.547063\pi$
$311$	−1615.91	−0.294629	−0.147315	+	0.989090i	$0.547063\pi$
$312$	0	0
$313$	−8479.33	−1.53125	−0.765623	−	0.643289i	$-0.777569\pi$
$313$	−8479.33	−1.53125	−0.765623	+	0.643289i	$0.777569\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6774.73	−1.20034	−0.600169	−	0.799873i	$-0.704900\pi$
$317$	−6774.73	−1.20034	−0.600169	+	0.799873i	$0.704900\pi$
$318$	0	0
$319$	−11049.2	−1.93929
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4988.44	−0.859332
$324$	0	0
$325$	−1534.00	−0.261818
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7038.57	1.17948
$330$	0	0
$331$	−9292.36	−1.54306	−0.771532	−	0.636191i	$-0.780509\pi$
$331$	−9292.36	−1.54306	−0.771532	+	0.636191i	$0.780509\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8552.80	1.39489
$336$	0	0
$337$	6563.78	1.06098	0.530492	−	0.847690i	$-0.322007\pi$
$337$	6563.78	1.06098	0.530492	+	0.847690i	$0.322007\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−7918.49	−1.25751
$342$	0	0
$343$	−4108.75	−0.646798
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3870.93	−0.598855	−0.299427	−	0.954119i	$-0.596796\pi$
$347$	−3870.93	−0.598855	−0.299427	+	0.954119i	$0.596796\pi$
$348$	0	0
$349$	−3474.57	−0.532922	−0.266461	−	0.963846i	$-0.585854\pi$
$349$	−3474.57	−0.532922	−0.266461	+	0.963846i	$0.585854\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4308.58	0.649639	0.324820	−	0.945776i	$-0.394696\pi$
$353$	4308.58	0.649639	0.324820	+	0.945776i	$0.394696\pi$
$354$	0	0
$355$	19395.7	2.89977
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8161.19	1.19981	0.599904	−	0.800072i	$-0.295205\pi$
$359$	8161.19	1.19981	0.599904	+	0.800072i	$0.295205\pi$
$360$	0	0
$361$	8938.68	1.30320
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18832.4	−2.70064
$366$	0	0
$367$	4427.66	0.629760	0.314880	−	0.949132i	$-0.398036\pi$
$367$	4427.66	0.629760	0.314880	+	0.949132i	$0.398036\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−994.045	−0.139106
$372$	0	0
$373$	−11278.5	−1.56562	−0.782812	−	0.622258i	$-0.786215\pi$
$373$	−11278.5	−1.56562	−0.782812	+	0.622258i	$0.786215\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1472.62	−0.201177
$378$	0	0
$379$	−709.683	−0.0961846	−0.0480923	−	0.998843i	$-0.515314\pi$
$379$	−709.683	−0.0961846	−0.0480923	+	0.998843i	$0.515314\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1233.69	0.164592	0.0822962	−	0.996608i	$-0.473775\pi$
$383$	1233.69	0.164592	0.0822962	+	0.996608i	$0.473775\pi$
$384$	0	0
$385$	22177.1	2.93571
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−6830.06	−0.890226	−0.445113	−	0.895474i	$-0.646836\pi$
$389$	−6830.06	−0.890226	−0.445113	+	0.895474i	$0.646836\pi$
$390$	0	0
$391$	3935.37	0.509003
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11106.9	1.41481
$396$	0	0
$397$	11289.7	1.42724	0.713618	−	0.700535i	$-0.247055\pi$
$397$	11289.7	1.42724	0.713618	+	0.700535i	$0.247055\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3055.59	0.380521	0.190261	−	0.981734i	$-0.439067\pi$
$401$	3055.59	0.380521	0.190261	+	0.981734i	$0.439067\pi$
$402$	0	0
$403$	−1055.37	−0.130451
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6748.08	0.821843
$408$	0	0
$409$	4089.01	0.494349	0.247175	−	0.968971i	$-0.420498\pi$
$409$	4089.01	0.494349	0.247175	+	0.968971i	$0.420498\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7264.79	0.865562
$414$	0	0
$415$	−291.471	−0.0344765
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15397.0	1.79520	0.897602	−	0.440806i	$-0.145307\pi$
$419$	15397.0	1.79520	0.897602	+	0.440806i	$0.145307\pi$
$420$	0	0
$421$	−1034.45	−0.119753	−0.0598766	−	0.998206i	$-0.519071\pi$
$421$	−1034.45	−0.119753	−0.0598766	+	0.998206i	$0.519071\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8508.40	−0.971101
$426$	0	0
$427$	−7264.79	−0.823344
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4943.86	0.552523	0.276261	−	0.961083i	$-0.410904\pi$
$431$	4943.86	0.552523	0.276261	+	0.961083i	$0.410904\pi$
$432$	0	0
$433$	337.202	0.0374247	0.0187124	−	0.999825i	$-0.494043\pi$
$433$	337.202	0.0374247	0.0187124	+	0.999825i	$0.494043\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12462.7	−1.36424
$438$	0	0
$439$	4493.93	0.488573	0.244286	−	0.969703i	$-0.421446\pi$
$439$	4493.93	0.488573	0.244286	+	0.969703i	$0.421446\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4292.26	0.460341	0.230171	−	0.973150i	$-0.426072\pi$
$443$	4292.26	0.460341	0.230171	+	0.973150i	$0.426072\pi$
$444$	0	0
$445$	7025.82	0.748440
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4167.96	0.438081	0.219040	−	0.975716i	$-0.429707\pi$
$449$	4167.96	0.438081	0.219040	+	0.975716i	$0.429707\pi$
$450$	0	0
$451$	−27190.4	−2.83890
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2955.73	0.304543
$456$	0	0
$457$	301.643	0.0308759	0.0154380	−	0.999881i	$-0.495086\pi$
$457$	301.643	0.0308759	0.0154380	+	0.999881i	$0.495086\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9611.88	0.971085	0.485542	−	0.874213i	$-0.338622\pi$
$461$	9611.88	0.971085	0.485542	+	0.874213i	$0.338622\pi$
$462$	0	0
$463$	−13251.0	−1.33008	−0.665041	−	0.746807i	$-0.731586\pi$
$463$	−13251.0	−1.33008	−0.665041	+	0.746807i	$0.731586\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4432.00	0.439161	0.219581	−	0.975594i	$-0.429531\pi$
$467$	4432.00	0.439161	0.219581	+	0.975594i	$0.429531\pi$
$468$	0	0
$469$	−10409.9	−1.02491
$470$	0	0
$471$	0	0
$472$	0	0
$473$	22210.5	2.15907
$474$	0	0
$475$	26944.9	2.60277
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9076.49	−0.865794	−0.432897	−	0.901443i	$-0.642509\pi$
$479$	−9076.49	−0.865794	−0.432897	+	0.901443i	$0.642509\pi$
$480$	0	0
$481$	899.378	0.0852559
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12143.1	−1.13688
$486$	0	0
$487$	−3343.89	−0.311142	−0.155571	−	0.987825i	$-0.549722\pi$
$487$	−3343.89	−0.311142	−0.155571	+	0.987825i	$0.549722\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	2423.73	0.222773	0.111386	−	0.993777i	$-0.464471\pi$
$491$	2423.73	0.222773	0.111386	+	0.993777i	$0.464471\pi$
$492$	0	0
$493$	−8167.95	−0.746179
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23607.1	−2.13063
$498$	0	0
$499$	811.819	0.0728296	0.0364148	−	0.999337i	$-0.488406\pi$
$499$	811.819	0.0728296	0.0364148	+	0.999337i	$0.488406\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18192.4	−1.61264	−0.806320	−	0.591479i	$-0.798544\pi$
$503$	−18192.4	−1.61264	−0.806320	+	0.591479i	$0.798544\pi$
$504$	0	0
$505$	−9183.65	−0.809242
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5645.44	0.491610	0.245805	−	0.969319i	$-0.420948\pi$
$509$	5645.44	0.491610	0.245805	+	0.969319i	$0.420948\pi$
$510$	0	0
$511$	22921.5	1.98432
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3625.88	0.310243
$516$	0	0
$517$	−16853.5	−1.43369
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12338.7	−1.03756	−0.518780	−	0.854908i	$-0.673614\pi$
$521$	−12338.7	−1.03756	−0.518780	+	0.854908i	$0.673614\pi$
$522$	0	0
$523$	−10609.8	−0.887062	−0.443531	−	0.896259i	$-0.646274\pi$
$523$	−10609.8	−0.887062	−0.443531	+	0.896259i	$0.646274\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5853.65	−0.483850
$528$	0	0
$529$	−2335.17	−0.191926
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3623.91	−0.294501
$534$	0	0
$535$	−6620.53	−0.535010
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−8577.07	−0.685419
$540$	0	0
$541$	4035.42	0.320696	0.160348	−	0.987061i	$-0.448738\pi$
$541$	4035.42	0.320696	0.160348	+	0.987061i	$0.448738\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	36286.8	2.85203
$546$	0	0
$547$	7407.45	0.579012	0.289506	−	0.957176i	$-0.406509\pi$
$547$	7407.45	0.579012	0.289506	+	0.957176i	$0.406509\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	25866.7	1.99993
$552$	0	0
$553$	−13518.6	−1.03954
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9500.77	0.722730	0.361365	−	0.932424i	$-0.382311\pi$
$557$	9500.77	0.722730	0.361365	+	0.932424i	$0.382311\pi$
$558$	0	0
$559$	2960.19	0.223976
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2700.26	−0.202136	−0.101068	−	0.994880i	$-0.532226\pi$
$563$	−2700.26	−0.202136	−0.101068	+	0.994880i	$0.532226\pi$
$564$	0	0
$565$	−12778.4	−0.951492
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15904.9	−1.17183	−0.585913	−	0.810374i	$-0.699264\pi$
$569$	−15904.9	−1.17183	−0.585913	+	0.810374i	$0.699264\pi$
$570$	0	0
$571$	18234.0	1.33638	0.668188	−	0.743992i	$-0.267070\pi$
$571$	18234.0	1.33638	0.668188	+	0.743992i	$0.267070\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−21256.7	−1.54168
$576$	0	0
$577$	4869.57	0.351339	0.175670	−	0.984449i	$-0.443791\pi$
$577$	4869.57	0.351339	0.175670	+	0.984449i	$0.443791\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	354.758	0.0253319
$582$	0	0
$583$	2380.19	0.169086
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1616.99	0.113697	0.0568485	−	0.998383i	$-0.481895\pi$
$587$	1616.99	0.113697	0.0568485	+	0.998383i	$0.481895\pi$
$588$	0	0
$589$	18537.7	1.29683
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8117.01	−0.562101	−0.281050	−	0.959693i	$-0.590683\pi$
$593$	−8117.01	−0.562101	−0.281050	+	0.959693i	$0.590683\pi$
$594$	0	0
$595$	16394.1	1.12957
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9536.40	−0.650495	−0.325248	−	0.945629i	$-0.605448\pi$
$599$	−9536.40	−0.650495	−0.325248	+	0.945629i	$0.605448\pi$
$600$	0	0
$601$	−16247.8	−1.10276	−0.551381	−	0.834253i	$-0.685899\pi$
$601$	−16247.8	−1.10276	−0.551381	+	0.834253i	$0.685899\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−28581.9	−1.92069
$606$	0	0
$607$	27725.7	1.85396	0.926980	−	0.375111i	$-0.122395\pi$
$607$	27725.7	1.85396	0.926980	+	0.375111i	$0.122395\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2246.22	−0.148727
$612$	0	0
$613$	927.190	0.0610911	0.0305456	−	0.999533i	$-0.490276\pi$
$613$	927.190	0.0610911	0.0305456	+	0.999533i	$0.490276\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18727.8	1.22196	0.610982	−	0.791644i	$-0.290775\pi$
$617$	18727.8	1.22196	0.610982	+	0.791644i	$0.290775\pi$
$618$	0	0
$619$	3210.22	0.208449	0.104224	−	0.994554i	$-0.466764\pi$
$619$	3210.22	0.208449	0.104224	+	0.994554i	$0.466764\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8551.33	−0.549922
$624$	0	0
$625$	3535.57	0.226276
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4988.44	0.316220
$630$	0	0
$631$	−11911.2	−0.751468	−0.375734	−	0.926728i	$-0.622609\pi$
$631$	−11911.2	−0.751468	−0.375734	+	0.926728i	$0.622609\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−48943.7	−3.05869
$636$	0	0
$637$	−1143.14	−0.0711036
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17232.6	−1.06185	−0.530924	−	0.847419i	$-0.678155\pi$
$641$	−17232.6	−1.06185	−0.530924	+	0.847419i	$0.678155\pi$
$642$	0	0
$643$	−12754.6	−0.782262	−0.391131	−	0.920335i	$-0.627916\pi$
$643$	−12754.6	−0.782262	−0.391131	+	0.920335i	$0.627916\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9441.73	−0.573714	−0.286857	−	0.957973i	$-0.592610\pi$
$647$	−9441.73	−0.573714	−0.286857	+	0.957973i	$0.592610\pi$
$648$	0	0
$649$	−17395.2	−1.05211
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5198.99	0.311565	0.155783	−	0.987791i	$-0.450210\pi$
$653$	5198.99	0.311565	0.155783	+	0.987791i	$0.450210\pi$
$654$	0	0
$655$	−11343.2	−0.676664
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−6508.01	−0.384698	−0.192349	−	0.981327i	$-0.561611\pi$
$659$	−6508.01	−0.384698	−0.192349	+	0.981327i	$0.561611\pi$
$660$	0	0
$661$	25280.2	1.48757	0.743785	−	0.668419i	$-0.233028\pi$
$661$	25280.2	1.48757	0.743785	+	0.668419i	$0.233028\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−51917.8	−3.02750
$666$	0	0
$667$	−20406.2	−1.18460
$668$	0	0
$669$	0	0
$670$	0	0
$671$	17395.2	1.00079
$672$	0	0
$673$	−5525.89	−0.316504	−0.158252	−	0.987399i	$-0.550586\pi$
$673$	−5525.89	−0.316504	−0.158252	+	0.987399i	$0.550586\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6293.21	0.357264	0.178632	−	0.983916i	$-0.442833\pi$
$677$	6293.21	0.357264	0.178632	+	0.983916i	$0.442833\pi$
$678$	0	0
$679$	14779.7	0.835335
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5675.91	0.317984	0.158992	−	0.987280i	$-0.449176\pi$
$683$	5675.91	0.317984	0.158992	+	0.987280i	$0.449176\pi$
$684$	0	0
$685$	−11298.2	−0.630190
$686$	0	0
$687$	0	0
$688$	0	0
$689$	317.229	0.0175406
$690$	0	0
$691$	−3617.79	−0.199171	−0.0995854	−	0.995029i	$-0.531752\pi$
$691$	−3617.79	−0.199171	−0.0995854	+	0.995029i	$0.531752\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−34989.1	−1.90966
$696$	0	0
$697$	−20100.2	−1.09232
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7938.43	0.427718	0.213859	−	0.976865i	$-0.431397\pi$
$701$	7938.43	0.427718	0.213859	+	0.976865i	$0.431397\pi$
$702$	0	0
$703$	−15797.7	−0.847540
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11177.7	0.594597
$708$	0	0
$709$	−25691.6	−1.36088	−0.680442	−	0.732802i	$-0.738212\pi$
$709$	−25691.6	−1.36088	−0.680442	+	0.732802i	$0.738212\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14624.3	−0.768142
$714$	0	0
$715$	−7077.35	−0.370179
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−36803.3	−1.90895	−0.954473	−	0.298298i	$-0.903581\pi$
$719$	−36803.3	−1.90895	−0.954473	+	0.298298i	$0.903581\pi$
$720$	0	0
$721$	−4413.16	−0.227954
$722$	0	0
$723$	0	0
$724$	0	0
$725$	44118.9	2.26005
$726$	0	0
$727$	28333.2	1.44542	0.722709	−	0.691152i	$-0.242897\pi$
$727$	28333.2	1.44542	0.722709	+	0.691152i	$0.242897\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16418.8	0.830742
$732$	0	0
$733$	−10767.9	−0.542592	−0.271296	−	0.962496i	$-0.587452\pi$
$733$	−10767.9	−0.542592	−0.271296	+	0.962496i	$0.587452\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24925.9	1.24580
$738$	0	0
$739$	−17301.4	−0.861221	−0.430610	−	0.902538i	$-0.641702\pi$
$739$	−17301.4	−0.861221	−0.430610	+	0.902538i	$0.641702\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24110.8	−1.19050	−0.595248	−	0.803542i	$-0.702946\pi$
$743$	−24110.8	−1.19050	−0.595248	+	0.803542i	$0.702946\pi$
$744$	0	0
$745$	17987.0	0.884555
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8058.04	0.393103
$750$	0	0
$751$	30052.8	1.46024	0.730121	−	0.683318i	$-0.239464\pi$
$751$	30052.8	1.46024	0.730121	+	0.683318i	$0.239464\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12584.8	−0.606633
$756$	0	0
$757$	25599.4	1.22910	0.614549	−	0.788879i	$-0.289338\pi$
$757$	25599.4	1.22910	0.614549	+	0.788879i	$0.289338\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	28904.5	1.37685	0.688427	−	0.725306i	$-0.258302\pi$
$761$	28904.5	1.37685	0.688427	+	0.725306i	$0.258302\pi$
$762$	0	0
$763$	−44165.7	−2.09555
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2318.41	−0.109143
$768$	0	0
$769$	−8756.13	−0.410603	−0.205302	−	0.978699i	$-0.565818\pi$
$769$	−8756.13	−0.410603	−0.205302	+	0.978699i	$0.565818\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−28289.8	−1.31632	−0.658159	−	0.752879i	$-0.728665\pi$
$773$	−28289.8	−1.31632	−0.658159	+	0.752879i	$0.728665\pi$
$774$	0	0
$775$	31618.3	1.46550
$776$	0	0
$777$	0	0
$778$	0	0
$779$	63654.4	2.92767
$780$	0	0
$781$	56526.0	2.58983
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9415.75	−0.428105
$786$	0	0
$787$	4859.74	0.220116	0.110058	−	0.993925i	$-0.464896\pi$
$787$	4859.74	0.220116	0.110058	+	0.993925i	$0.464896\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	15553.0	0.699117
$792$	0	0
$793$	2318.41	0.103820
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17361.7	−0.771623	−0.385811	−	0.922578i	$-0.626078\pi$
$797$	−17361.7	−0.771623	−0.385811	+	0.922578i	$0.626078\pi$
$798$	0	0
$799$	−12458.8	−0.551638
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−54884.2	−2.41198
$804$	0	0
$805$	40957.8	1.79326
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24475.3	1.06367	0.531833	−	0.846849i	$-0.321503\pi$
$809$	24475.3	1.06367	0.531833	+	0.846849i	$0.321503\pi$
$810$	0	0
$811$	19875.4	0.860566	0.430283	−	0.902694i	$-0.358414\pi$
$811$	19875.4	0.860566	0.430283	+	0.902694i	$0.358414\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	44691.4	1.92083
$816$	0	0
$817$	−51996.1	−2.22658
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21682.1	−0.921693	−0.460846	−	0.887480i	$-0.652454\pi$
$821$	−21682.1	−0.921693	−0.460846	+	0.887480i	$0.652454\pi$
$822$	0	0
$823$	6698.17	0.283698	0.141849	−	0.989888i	$-0.454695\pi$
$823$	6698.17	0.283698	0.141849	+	0.989888i	$0.454695\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3390.85	0.142577	0.0712886	−	0.997456i	$-0.477289\pi$
$827$	3390.85	0.142577	0.0712886	+	0.997456i	$0.477289\pi$
$828$	0	0
$829$	−40093.4	−1.67974	−0.839869	−	0.542789i	$-0.817368\pi$
$829$	−40093.4	−1.67974	−0.839869	+	0.542789i	$0.817368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6340.50	−0.263728
$834$	0	0
$835$	6214.40	0.257555
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6172.12	0.253975	0.126988	−	0.991904i	$-0.459469\pi$
$839$	6172.12	0.253975	0.126988	+	0.991904i	$0.459469\pi$
$840$	0	0
$841$	17964.6	0.736585
$842$	0	0
$843$	0	0
$844$	0	0
$845$	39530.3	1.60933
$846$	0	0
$847$	34787.8	1.41124
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12462.7	0.502018
$852$	0	0
$853$	−276.632	−0.0111040	−0.00555198	−	0.999985i	$-0.501767\pi$
$853$	−276.632	−0.0111040	−0.00555198	+	0.999985i	$0.501767\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3704.41	−0.147655	−0.0738274	−	0.997271i	$-0.523521\pi$
$857$	−3704.41	−0.147655	−0.0738274	+	0.997271i	$0.523521\pi$
$858$	0	0
$859$	26915.5	1.06909	0.534544	−	0.845141i	$-0.320484\pi$
$859$	26915.5	1.06909	0.534544	+	0.845141i	$0.320484\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23623.9	0.931828	0.465914	−	0.884830i	$-0.345726\pi$
$863$	23623.9	0.931828	0.465914	+	0.884830i	$0.345726\pi$
$864$	0	0
$865$	48787.3	1.91771
$866$	0	0
$867$	0	0
$868$	0	0
$869$	32369.5	1.26359
$870$	0	0
$871$	3322.10	0.129236
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−36918.9	−1.42638
$876$	0	0
$877$	14094.0	0.542667	0.271333	−	0.962485i	$-0.412535\pi$
$877$	14094.0	0.542667	0.271333	+	0.962485i	$0.412535\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18967.3	−0.725341	−0.362671	−	0.931917i	$-0.618135\pi$
$881$	−18967.3	−0.725341	−0.362671	+	0.931917i	$0.618135\pi$
$882$	0	0
$883$	32886.0	1.25334	0.626672	−	0.779283i	$-0.284417\pi$
$883$	32886.0	1.25334	0.626672	+	0.779283i	$0.284417\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27945.6	1.05786	0.528929	−	0.848666i	$-0.322594\pi$
$887$	27945.6	1.05786	0.528929	+	0.848666i	$0.322594\pi$
$888$	0	0
$889$	59570.8	2.24740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	39455.1	1.47852
$894$	0	0
$895$	−53558.6	−2.00030
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30353.1	1.12607
$900$	0	0
$901$	1759.53	0.0650592
$902$	0	0
$903$	0	0
$904$	0	0
$905$	67845.2	2.49199
$906$	0	0
$907$	−5544.89	−0.202993	−0.101497	−	0.994836i	$-0.532363\pi$
$907$	−5544.89	−0.202993	−0.101497	+	0.994836i	$0.532363\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19638.9	−0.714233	−0.357116	−	0.934060i	$-0.616240\pi$
$911$	−19638.9	−0.714233	−0.357116	+	0.934060i	$0.616240\pi$
$912$	0	0
$913$	−849.451	−0.0307916
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13806.1	0.497184
$918$	0	0
$919$	22128.7	0.794297	0.397149	−	0.917754i	$-0.370000\pi$
$919$	22128.7	0.794297	0.397149	+	0.917754i	$0.370000\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7533.72	0.268663
$924$	0	0
$925$	−26944.9	−0.957775
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17599.3	−0.621544	−0.310772	−	0.950484i	$-0.600588\pi$
$929$	−17599.3	−0.621544	−0.310772	+	0.950484i	$0.600588\pi$
$930$	0	0
$931$	20079.5	0.706850
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−39254.9	−1.37302
$936$	0	0
$937$	441.299	0.0153859	0.00769297	−	0.999970i	$-0.497551\pi$
$937$	441.299	0.0153859	0.00769297	+	0.999970i	$0.497551\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5408.01	−0.187350	−0.0936749	−	0.995603i	$-0.529861\pi$
$941$	−5408.01	−0.187350	−0.0936749	+	0.995603i	$0.529861\pi$
$942$	0	0
$943$	−50216.8	−1.73413
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34237.1	1.17482	0.587410	−	0.809289i	$-0.300148\pi$
$947$	34237.1	1.17482	0.587410	+	0.809289i	$0.300148\pi$
$948$	0	0
$949$	−7314.92	−0.250213
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21410.7	−0.727766	−0.363883	−	0.931445i	$-0.618549\pi$
$953$	−21410.7	−0.727766	−0.363883	+	0.931445i	$0.618549\pi$
$954$	0	0
$955$	23572.2	0.798722
$956$	0	0
$957$	0	0
$958$	0	0
$959$	13751.3	0.463038
$960$	0	0
$961$	−8038.09	−0.269816
$962$	0	0
$963$	0	0
$964$	0	0
$965$	89105.3	2.97243
$966$	0	0
$967$	−53874.6	−1.79161	−0.895806	−	0.444445i	$-0.853401\pi$
$967$	−53874.6	−1.79161	−0.895806	+	0.444445i	$0.853401\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42901.5	−1.41789	−0.708947	−	0.705262i	$-0.750829\pi$
$971$	−42901.5	−1.41789	−0.708947	+	0.705262i	$0.750829\pi$
$972$	0	0
$973$	42586.2	1.40314
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58636.5	−1.92011	−0.960055	−	0.279812i	$-0.909728\pi$
$977$	−58636.5	−1.92011	−0.960055	+	0.279812i	$0.909728\pi$
$978$	0	0
$979$	20475.7	0.668444
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25296.7	0.820793	0.410396	−	0.911907i	$-0.365390\pi$
$983$	25296.7	0.820793	0.410396	+	0.911907i	$0.365390\pi$
$984$	0	0
$985$	52859.0	1.70988
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41019.6	1.31885
$990$	0	0
$991$	10605.2	0.339944	0.169972	−	0.985449i	$-0.445632\pi$
$991$	10605.2	0.339944	0.169972	+	0.985449i	$0.445632\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12015.7	−0.382839
$996$	0	0
$997$	−5770.19	−0.183294	−0.0916469	−	0.995792i	$-0.529213\pi$
$997$	−5770.19	−0.183294	−0.0916469	+	0.995792i	$0.529213\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.4.a.t.1.1		2
3.2	odd	2		768.4.a.j.1.2		2
4.3	odd	2		2304.4.a.s.1.1		2
8.3	odd	2		2304.4.a.bp.1.2		2
8.5	even	2		2304.4.a.bq.1.2		2
12.11	even	2		768.4.a.p.1.2		2
16.3	odd	4		1152.4.d.o.577.4		4
16.5	even	4		1152.4.d.i.577.1		4
16.11	odd	4		1152.4.d.o.577.1		4
16.13	even	4		1152.4.d.i.577.4		4
24.5	odd	2		768.4.a.k.1.1		2
24.11	even	2		768.4.a.e.1.1		2
48.5	odd	4		384.4.d.c.193.4	yes	4
48.11	even	4		384.4.d.e.193.2	yes	4
48.29	odd	4		384.4.d.c.193.1	✓	4
48.35	even	4		384.4.d.e.193.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.d.c.193.1	✓	4	48.29	odd	4
384.4.d.c.193.4	yes	4	48.5	odd	4
384.4.d.e.193.2	yes	4	48.11	even	4
384.4.d.e.193.3	yes	4	48.35	even	4
768.4.a.e.1.1		2	24.11	even	2
768.4.a.j.1.2		2	3.2	odd	2
768.4.a.k.1.1		2	24.5	odd	2
768.4.a.p.1.2		2	12.11	even	2
1152.4.d.i.577.1		4	16.5	even	4
1152.4.d.i.577.4		4	16.13	even	4
1152.4.d.o.577.1		4	16.11	odd	4
1152.4.d.o.577.4		4	16.3	odd	4
2304.4.a.s.1.1		2	4.3	odd	2
2304.4.a.t.1.1		2	1.1	even	1	trivial
2304.4.a.bp.1.2		2	8.3	odd	2
2304.4.a.bq.1.2		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

\(f(q)\)	\(=\)	\(q-18.4222 q^{5} +22.4222 q^{7} +O(q^{10})\)	\(q-18.4222 q^{5} +22.4222 q^{7} -53.6888 q^{11} -7.15559 q^{13} -39.6888 q^{17} +125.689 q^{19} -99.1556 q^{23} +214.378 q^{25} +205.800 q^{29} +147.489 q^{31} -413.066 q^{35} -125.689 q^{37} +506.444 q^{41} -413.689 q^{43} +313.911 q^{47} +159.755 q^{49} -44.3331 q^{53} +989.066 q^{55} +324.000 q^{59} -324.000 q^{61} +131.822 q^{65} -464.266 q^{67} -1052.84 q^{71} +1022.27 q^{73} -1203.82 q^{77} -602.910 q^{79} +15.8217 q^{83} +731.156 q^{85} -381.378 q^{89} -160.444 q^{91} -2315.47 q^{95} +659.154 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{5} + 16 q^{7}+O(q^{10}) \) 2 * q - 8 * q^5 + 16 * q^7	\( 2 q - 8 q^{5} + 16 q^{7} + 8 q^{11} - 72 q^{13} + 36 q^{17} + 136 q^{19} - 256 q^{23} + 198 q^{25} + 152 q^{29} - 80 q^{31} - 480 q^{35} - 136 q^{37} + 436 q^{41} - 712 q^{43} + 224 q^{47} - 142 q^{49} + 344 q^{53} + 1632 q^{55} + 648 q^{59} - 648 q^{61} - 544 q^{65} + 456 q^{67} - 2048 q^{71} + 660 q^{73} - 1600 q^{77} + 496 q^{79} - 776 q^{83} + 1520 q^{85} - 532 q^{89} + 256 q^{91} - 2208 q^{95} - 1220 q^{97}+O(q^{100}) \) 2 * q - 8 * q^5 + 16 * q^7 + 8 * q^11 - 72 * q^13 + 36 * q^17 + 136 * q^19 - 256 * q^23 + 198 * q^25 + 152 * q^29 - 80 * q^31 - 480 * q^35 - 136 * q^37 + 436 * q^41 - 712 * q^43 + 224 * q^47 - 142 * q^49 + 344 * q^53 + 1632 * q^55 + 648 * q^59 - 648 * q^61 - 544 * q^65 + 456 * q^67 - 2048 * q^71 + 660 * q^73 - 1600 * q^77 + 496 * q^79 - 776 * q^83 + 1520 * q^85 - 532 * q^89 + 256 * q^91 - 2208 * q^95 - 1220 * q^97

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